Event Type:

Department Colloquium

Date/Time:

Friday, April 27, 2007 - 08:00

Location:

Kidd 364

Guest Speaker:

Institution:

U. Washington

Abstract:

In classical probability, the Gaussian, Chi-square, and Beta are three of the most studied distributions, with wide applicability. In the last century, matrix equivalents to these three distributions have emerged from nuclear physics (Gaussian ensembles) and multivariate statistics (Wishart and MANOVA ensembles). Their eigenvalue statistics have been studied in depth for three values of a parameter (β = 1, 2 and 4) which defines the "threefold way" and can be thought of as a counting tool for their real, complex, or quaternion entries.

The re-examination of the Selberg integral formula, in the late `80s, has brought the advent of general β-ensembles, which subsume the classical cases, and for which the Boltzmann parameter β acts as an inverse temperature. Their eigenvalue statistics interpolate between the isolated instances 1, 2, and 4, offering a "behind the scenes" perspective.

With the discovery of matrix models for the general β-ensembles in the early 00's, we have entered a new stage in the understanding of the complex phenomena that lie beneath the threefold way. While the β = 1,2 and 4 cases are and will always be special, we can now argue that the future of the classical ensembles is written in terms of a continuous β parameter.